Suppose a group of people are voting on a particular value, for example the level of taxation. The obvious way to do this would be to average everyone's prefered value. But this is easy to distort: say someone wanted 10% taxation, but knew other people would be voting for 1%. Then this person should obviously vote 100% taxation, so as to bump up the result as high as possible. Averaging obviously won't do.

Suppose instead that the vote is like a "tug-of-war" in which everyone is equally strong. Each person can pull the value up, pull the value down, or do nothing. Supposing each person has a preferred value, their strategy is simple: pull up if the value is too low, pull down if it's too high, or do nothing if it's just right. For convenience, let's assume an odd number of voters. The equilibreum state of this tug-of-war is then fairly obviously one where one person is doing nothing and the remainder are evenly split between pulling up and down. In other words, the median of the preferred values.

Mr. 10% now gains no advantage by saying he wants preffers 100%, he's just another pull-upper. So he is best off saying he wants 10%. This is much nicer. In general, the median is considered to be a more robust measure of central tendancy than the mean.

Can this be generalized to higher dimensions, as in voting on more than one thing at once? I am less sure of this, but it could be useful. Again let's suppose a tug-of-war, but this time people can pull in any direction, though all are still equally strong.

For example, suppose you were voting on the level of spending for medicare and social welfare. If you were really keen on social welfare you might decide to be ambivalent about medicare in order to maximize your influence on welfare.

It's not quite clear to me how people would behave in this game though. The example shows that people will not simply pull straight towards their preferred outcome, they may try to achieve something in one dimension first, then if happy with that fix the other dimension. So it is not as clear what the equilibreum is.

One way of looking at this is to say that each person gets to specify a function they want maximized (like a "utility" function), with the condition that their function must be convex and that its gradient must at no point exceed a certain value. If we then add together all these functions, there will be a unique maximum point or (hopefully small) plateau. I think this is equivalent to the tug-of-war game. But it's a bit mathematical and abstruse to use in practice without some further thought.

Applications:

Proportional representation with more than two parties (as in the Australian senate).

Preferential voting?

A final thought: politics in the media tends to become polarized. People advocate an extreme view in the hope of achieving a more moderate objective. Sound familiar? Maybe instead of extremism, political groups should talk (forcefully) about which direction they want to pull things.